96 
may be both evenly distant; taking one of them as origin, 
it is associated with 5 mediates, so that there is one type 
only. Or both oddly distant; here there are two cases, 
according as the distances are 11, 33 or 13, 13. Or one oddly 
and one evenly distant; the latter is any one of the four 
remaining mediates, and then the former is distant 1 or 3 
from it ; 2 types. If the two pairs of obverses be oddly dis- 
tant, they form an aggregate which is related in the same 
way to all the remaining 12 marks; viz. any one of these 
being taken as origin, we have a pair of mediates and a 
proximate with its obverse ultimate. The thing to be 
considered, therefore, is the distance between the two marks 
to be added, which may be 1, 2 or 3, and each in two ways; 
6 types. 
A six-fold statement with 3 pairs of obverses is one of two 
types only; viz. these are all evenly distant, when they are 
the mediates to one origin, or two evenly distant and one 
oddly distant from both of them. 
13. A pure seven-fold statement must consist of a group 
and a triad ; for it must contain a triad, by the same reason- 
ing by which this was proved for a five-fold statement, and 
then either all the other four marks are oddly distant from 
this, and so form a group by themselves; or else one of 
them is evenly distant from the triad and so forms a group 
with it. If the group is proper, being the proximates to a 
certain origin, the triad must consist of two mediates and 
either the origin, the obverse or another mediate, and in the 
latter case the 3 mediates are distant 111 or 333 from some 
proximate ; 4 types. If the group is improper, the triad is 
either all origins or all obverses or two origins and an obverse 
or an origin and two obverses ; 4 types. In all 8 types of 
pure seven-fold statement. 
14. A seven-fold statement with one pair of obverses must 
consist either of four marks evenly distant from one another 
and three oddly distant from them ; or of five marks evenly 
